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Concept
Coxeter–Dynkin diagram
Summary
In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes). It describes a kaleidoscopic construction: each graph "node" represents a mirror (domain facet) and the label attached to a branch encodes the dihedral angle order between two mirrors (on a domain ridge), that is, the amount by which the angle between the reflective planes can be multiplied to get 180 degrees. An unlabeled branch implicitly represents order-3 (60 degrees), and each pair of nodes that is not connected by a branch at all (such as non-adjacent nodes) represents a pair of mirrors at order-2 (90 degrees).
Each diagram represents a Coxeter group, and Coxeter groups are classified by their associated diagrams.
Dynkin diagrams are closely related objects, which differ from Coxeter diagrams in two respects: firstly, branches labeled "4" or greater are directed, while Coxeter diagrams are undirected; secondly, Dynkin diagrams must satisfy an additional (crystallographic) restriction, namely that the only allowed branch labels are 2, 3, 4, and 6. Dynkin diagrams correspond to and are used to classify root systems and therefore semisimple Lie algebras.
Branches of a Coxeter–Dynkin diagram are labeled with a rational number p, representing a dihedral angle of 180°/p. When p = 2 the angle is 90° and the mirrors have no interaction, so the branch can be omitted from the diagram. If a branch is unlabeled, it is assumed to have p = 3, representing an angle of 60°. Two parallel mirrors have a branch marked with "∞". In principle, n mirrors can be represented by a complete graph in which all n(n − 1) / 2 branches are drawn. In practice, nearly all interesting configurations of mirrors include a number of right angles, so the corresponding branches are omitted.
Diagrams can be labeled by their graph structure. The first forms studied by Ludwig Schläfli are the orthoschemes which have linear graphs that generate regular polytopes and regular honeycombs.
Official source
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